Reconstructed by a model.
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2 Two models for gene expression¶
It is well-known that the common two-state model of gene expression [11] gives rise to a negative binomial (NB) distribution of mRNA counts in the short-burst limit [12]. However, a recent study shows that constitutive transcription in a cell population with a gamma-distributed production rate parameter also yields a negative binomial distribution of mRNA counts [13]. Although there are both experimental and theoretical arguments favoring a bursting model for eukaryotic transcription [14–17] – current theories posit that superstructure modifications are responsible for occlusion and exposure of the gene locus [18, 19] – a comprehensive model should account for all relevant sources of noise, as well as provide both a quantitative and qualitative understanding of their effects.
A current limitation of existing models is that processes downstream of eukaryotic mRNA production, such as export and/or splicing processes [20, 21], are generally ignored. However, promising new technologies and experiments, based on fluorescence [22, 23] and sequencing [24] methods, can distinguish nascent from mature mRNA, thus providing essential data for studying model of increasing complexity. In particular, the maturation of these methods and the availability of resulting multimodal data naturally suggests the potential of fitting otherwise poorly identifiable models [9].
As a first step, and to gain a qualitative understanding of the effects of intrinsic and extrinsic noise in the context of downstream processing, we compare solutions of two simple two-stage models of transcription that include downstream processing at steady state. Both models assume that nascent mRNA (unspliced or pre-mRNA) is converted to mature mRNA (spliced mRNA) after an exponentially-distributed delay, corresponding to splicing. This is followed by another exponentially-distributed delay that models the mature mRNA being degraded. The splicing rate \(\beta\) and degradation rate \(\gamma\) are deterministic. The gene locus dynamics are modeled by either bursts, with stochastic burst size \(B \sim Geom(b)\) and deterministic burst initiation frequency \(k_i\) [20], or constitutive, with stochastic but constant transcription rate \(K \sim Gamma(\alpha,\eta)\). The model parametrizations are illustrated in Figure 1. We calculate lower moments and cross-moments, and show how these can be used to differentiate between distributions and statistics resulting from the two models.
Figure 1: (a) Schema of the intrinsic noise model (\(k_i\): burst frequency; \(B\): burst size drawn from a geometric distribution; \(\beta\): pre-mRNA splicing rate; \(\gamma\): mRNA degradation rate. Uniform shade of green indicates identical parameter values across all cells). (b) Schema of the extrinsic noise model (\(K\): transcription rate; \(\beta\): pre-mRNA splicing rate; \(\gamma\): mRNA degradation rate. Different shades of green indicate different values of \(K\) across cells).
Schema: (a) Intrinsic noise model: Gene \(\xrightarrow{k_i}\) (burst \(B\times\)) Pre-mRNA \(\xrightarrow{\beta}\) mRNA \(\xrightarrow{\gamma} \emptyset\). (b) Extrinsic noise model: Gene \(\xrightarrow{K}\) Pre-mRNA \(\xrightarrow{\beta}\) mRNA \(\xrightarrow{\gamma} \emptyset\).